Suppose the prevalence of people with certain cancer in a give year is 0.015. In addition suppose that the diagnostic test for this cancer is know to have a false positive rate of 0.05 and a false negative rate of 0.10 . Suppose a person is aware of the false positive and false negative rates for this exam. So when he gets a postitive result he takes the exam again

a) if the second test is also positive what are his chances of actually having cancer
b) if the second test is negative what are his chances and should he get another test?

First calculate the probabilities for all possible outcomes. There are two kinds of people with and without cancer, the first test can have two outcomes positive and negative, and the second test has two outcomes. So, there are 8 possible outcomes.
So, the probability that a person has cancer and the two tests are positive is .015*.90*.90 = .01215. Repeat for the 7 other possibilities.
The probability that a person doesnt have cancer and the two tests are (false) positives is .985*.05*.05 =

First calculate the probabilities for all possible outcomes. There are two kinds of people with and without cancer, the first test can have two outcomes positive and negative, and the second test has two outcomes. So, there are 8 possible outcomes.
So, the probability that a person has cancer and the two tests are positive is .015*.90*.90 = .01215. Repeat for the 7 other possibilities.
The probability that a person doesnt have cancer and the two tests are (false) positives is .985*.05*.05 = .0024625

So, a) conditional on two positive tests, the likelihood of having cancer is .01215/(.01215+.0024625) = .831

b) take it from here.

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